Mathematical modeling of natural resource and human interaction: Applications to the harvesting of Pacific Yew for cancer treatment

Figure 1.  Nullclines $ nf_1(P) $ and $ nf_2(P) $ for Model (1), illustrating the existence of no interior equilibria (subfigures a and b), one interior equilibrium (subfigures c and d, $ \mathbf{E_1} $), and two interior equilibria (subfigure e, $ \mathbf{E_1} $ and $ \mathbf{E_2} $). The orange dot marks the intersection(s) of the nullclines, indicating the number of interior equilibria. The following parameter values were used: $ r_P = 0.7 $, $ r_C = 0.3 $, $ K_P = 10 $, and $ q = 0.1 $

Figure 2.  One parameter bifurcation for Model (1) as a function of the bifurcation parameter $ \alpha $ (resource's removal rate). Figures (a)-(c) are an example of the backward bifurcation that Model (1) undergoes at $ \alpha = \frac{r_P}{K_C} $. Parameter values used in figures (a)-(c) are: $ r_P = 0.7 $, $ r_C = 3.4 $, $ K_P = 9.5 $, $ K_C = 1 $, $ \beta = 0.4 $, $ q = 0.5 $, and $ \mu_R = 0.006 $. Figures (d)-(f) are a simulated example when Model (1) has a cancer-free population and we have natural resource persistance. Parameter values used in figures (d)-(f) are: $ r_P = 2.6 $, $ r_C = 0.9 $, $ K_P = 2 $, $ K_C = 0.9 $, $ \beta = 1.5 $, $ q = 1 $, and $ \mu_R = 0.006 $. The diagram shows the boundary and interior equilibria of the system, where blue represents stable equilibria and green indicates saddle points

Figure 3.  One parameter bifurcation for Model (1) as a function of the bifurcation parameter $ \beta $ (effort needed to harvest the natural resource and process treatment). This is an example of the forward bifurcation that Model (1) undergoes at $ \beta = \frac{q \alpha}{r_C}-\frac{1}{K_P} $. Parameter values used are: $ r_P = 0.42447 $, $ r_C = 0.043 $, $ K_P = 10 $, $ K_C = 5 $, $ \alpha = 0.6 $, $ \beta = 5 $, $ q = 0.1 $, and $ \mu_R = 0.006 $. The diagram shows the boundary and interior equilibria of the system, where blue represents stable equilibria and green indicates saddle points

Figure 4.  One parameter bifurcation for Model (1) as a function of the bifurcation parameter $ r_C $ (cancer patients max growth rate). Parameter values used are: $ r_P = 0.7 $, $ K_P = 9.5 $, $ K_C = 1.6 $, $ \alpha = 3.4 $, $ \beta = 0.4 $, $ q = 0.5 $, and $ \mu_R = 0.006 $. The diagram shows the interior equilibria of the system, where blue represents stable equilibria and green indicates saddle points

Figure 5.  One parameter bifurcation for Model (1) as a function of the bifurcation parameter $ r_P $ (resource maximum growth rate). Parameter values used are: $ r_C = 0.043 $, $ K_P = 10 $, $ K_C = 5 $, $ \alpha = 0.6 $, $ \beta = 5 $, $ q = 0.1 $, and $ \mu_R = 0.006 $. The diagram shows the interior equilibria of the system, where blue represents stable equilibria and green indicates saddle points

Figure 6.  One parameter bifurcation for Model (1) as a function of the bifurcation parameter $ q $ (conversion of cancer patients to recovered individuals due to treatment derived from natural resource). Parameter values used are: $ r_P = 0.42447 $, $ r_C = 0.043 $, $ K_P = 10 $, $ K_C = 5 $, $ \alpha = 0.6 $, $ \beta = 5 $, and $ \mu_R = 0.006 $. The diagram shows the interior equilibria of the system, where blue represents stable equilibria and green indicates saddle points

Figure 7.  Time series of the sum of lung, ovarian and breast cancer patients with the prevalence of cancer patients. All data was obtained from the Global Health Data Exchange [39]. We also show the best fit our model to the available data. The grey shaded section represents the first sectional fitting, while the green shaded section represents the second sectional fitting

Figure 8.  Figure (a) is a 3D bifurcation for parameters $ \alpha $, $ \beta $ and $ q $. Figure (b) is a 2D bifurcation for parameters $ \alpha $ and $ \beta $. The red and black markers represent the change of two and one interior equilibria, respectively. Figure (c) and (d) is a 1D bifurcation with respect to parameter $ \alpha $. This diagram shows the interior equilibria of the system, where blue represents stable equilibria and green indicates saddle points. Any empty area in each figure indicates no interior equilibrium in the system